#27393589

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1991 ⋊ C₂
Sylow	C₅₃ ⋊ #21836

Subgroups

	subgroup	G/H
center	C₂³	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1991	C₂
socle	C₂³×C₅₃	C₂²×D₄
prime radical	C₂³×D₅₃	C₂×D₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#1991	D₅₃
hypocenter	C₅₃	#21836
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1991

1042 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 89
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 4 generators on 89 points:

(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27)(28,29)(30,31)(32,33)(34,35)(36,37)(38,39)(40,41)(42,43)(44,45)(46,47)(48,49)(50,51)(52,53)(59,65)(63,66)(64,73)(71,72);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(58,61,62,70)(59,64,65,73)(60,67,68,69)(63,71,66,72)(75,80)(77,85)(79,86)(81,82)(83,89)(87,88);(58,60,62,68)(59,63,65,66)(61,69,70,67)(64,72,73,71)(74,76,78,84)(75,79,82,88)(77,83,85,89)(80,86,81,87);(54,55,56,57)(58,59)(60,66)(61,64)(62,65)(63,68)(67,72)(69,71)(70,73)(74,75)(76,79)(77,80)(78,82)(81,85)(83,86)(84,88)(87,89)

Polycyclic presentation:

2^8,53|a^2=g,b^2=eh,c^2=e,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=ce,f^c=fh,i^d=i^52

Conjugacy classes & character table

1208 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 179065);